Definite Integration

As the number of rectangles increased, the approximation of the area under the curve approaches a value.

Definition Note: The function f(x) must be continuous on the interval [a, b].

The numbers on the integral sign are called the limits of integration is a definite integral The numbers on the integral sign are called the limits of integration

Evaluating the Definite Integral The definite integration results in a value. e.g.1 Find the indefinite integral but omit C

Evaluating the Definite Integral The definite integration results in a value. e.g.1 Draw square brackets and hang the limits on the end

Evaluating the Definite Integral The definite integration results in a value. e.g.1 Replace x with the top limit the bottom limit

Evaluating the Definite Integral The definite integration results in a value. e.g.1 Subtract and evaluate

Evaluating the Definite Integral The definite integration results in a value. e.g.1 So,

SUMMARY The method for evaluating the definite integral is: Find the indefinite integral but omit C Draw square brackets and hang the limits on the end Replace x with the top limit the bottom limit Subtract and evaluate

Evaluating the Definite Integral e.g. 2 Find Solution: Indefinite integral but no C

Evaluating the Definite Integral e.g. 2 Find Solution: Substitute for x: top limit minus bottom limit Simplify

Evaluating the Definite Integral In this example, if we can’t use a calculator. We must be very careful with the signs

Examples 1. Find 2. Find

Examples

Examples

Examples

Area & Integration

Area & Integration

Area & Integration

Area & Integration

Area & Integration

Area & Integration